Asymptotic Results on the Spectral Radius and the Diameter of Graphs

2000 Mathematics Subject Classification: 05C50, 05E99;graphs;spectral radius;diameter;limit points;quipus

A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

AMS classifications: 05C50, 05E99;graphs;spectral radius;diameter;bound;degree/diameter

Asymptotic results on the spectral radius and the diameter of graphs

AbstractWe study graphs with spectral radius at most 322 and refine results by Woo and Neumaier [R. Woo, A. Neumaier, On graphs whose spectral radius is bounded by 322, Graphs Combinatorics 23 (2007) 713–726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E. R. Van Dam, R. E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419] and show that the conjecture is false for the other instances

Matchings in regular graphs from eigenvalues

AbstractLet G be a connected k-regular graph of order n. We find a best upper bound (in terms of k) on the third largest eigenvalue that is sufficient to guarantee that G has a perfect matching when n is even, and a matching of order n−1 when n is odd. We also examine how other eigenvalues affect the size of matchings in G

A lower bound for the spectral radius of graphs with fixed diameter

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem

A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

AMS classifications: 05C50, 05E99